Properties

Label 177.10.a.c.1.15
Level $177$
Weight $10$
Character 177.1
Self dual yes
Analytic conductor $91.161$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(91.1613430010\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+18.6113 q^{2} -81.0000 q^{3} -165.619 q^{4} +1492.75 q^{5} -1507.52 q^{6} +11801.9 q^{7} -12611.4 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+18.6113 q^{2} -81.0000 q^{3} -165.619 q^{4} +1492.75 q^{5} -1507.52 q^{6} +11801.9 q^{7} -12611.4 q^{8} +6561.00 q^{9} +27782.1 q^{10} +54115.8 q^{11} +13415.1 q^{12} +176081. q^{13} +219649. q^{14} -120913. q^{15} -149917. q^{16} +108748. q^{17} +122109. q^{18} +54464.9 q^{19} -247228. q^{20} -955953. q^{21} +1.00717e6 q^{22} -2.05205e6 q^{23} +1.02152e6 q^{24} +275183. q^{25} +3.27711e6 q^{26} -531441. q^{27} -1.95462e6 q^{28} -3.15094e6 q^{29} -2.25035e6 q^{30} +5.67861e6 q^{31} +3.66687e6 q^{32} -4.38338e6 q^{33} +2.02394e6 q^{34} +1.76173e7 q^{35} -1.08663e6 q^{36} +1.48233e7 q^{37} +1.01366e6 q^{38} -1.42626e7 q^{39} -1.88257e7 q^{40} -2.73520e7 q^{41} -1.77915e7 q^{42} +1.96666e7 q^{43} -8.96260e6 q^{44} +9.79394e6 q^{45} -3.81914e7 q^{46} -1.60710e7 q^{47} +1.21433e7 q^{48} +9.89308e7 q^{49} +5.12152e6 q^{50} -8.80857e6 q^{51} -2.91624e7 q^{52} -4.91083e7 q^{53} -9.89081e6 q^{54} +8.07814e7 q^{55} -1.48838e8 q^{56} -4.41166e6 q^{57} -5.86430e7 q^{58} +1.21174e7 q^{59} +2.00255e7 q^{60} +9.09669e7 q^{61} +1.05686e8 q^{62} +7.74322e7 q^{63} +1.45003e8 q^{64} +2.62846e8 q^{65} -8.15804e7 q^{66} -1.83319e7 q^{67} -1.80107e7 q^{68} +1.66216e8 q^{69} +3.27881e8 q^{70} -3.08979e8 q^{71} -8.27433e7 q^{72} +1.21812e7 q^{73} +2.75880e8 q^{74} -2.22898e7 q^{75} -9.02043e6 q^{76} +6.38668e8 q^{77} -2.65446e8 q^{78} -2.20744e8 q^{79} -2.23789e8 q^{80} +4.30467e7 q^{81} -5.09056e8 q^{82} +4.48981e8 q^{83} +1.58324e8 q^{84} +1.62333e8 q^{85} +3.66022e8 q^{86} +2.55226e8 q^{87} -6.82475e8 q^{88} +7.18855e8 q^{89} +1.82278e8 q^{90} +2.07809e9 q^{91} +3.39859e8 q^{92} -4.59968e8 q^{93} -2.99103e8 q^{94} +8.13026e7 q^{95} -2.97016e8 q^{96} -1.57982e9 q^{97} +1.84123e9 q^{98} +3.55054e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22q + 36q^{2} - 1782q^{3} + 5718q^{4} + 808q^{5} - 2916q^{6} + 21249q^{7} + 9435q^{8} + 144342q^{9} + O(q^{10}) \) \( 22q + 36q^{2} - 1782q^{3} + 5718q^{4} + 808q^{5} - 2916q^{6} + 21249q^{7} + 9435q^{8} + 144342q^{9} + 68441q^{10} - 68033q^{11} - 463158q^{12} + 283817q^{13} + 80285q^{14} - 65448q^{15} + 1067674q^{16} + 436893q^{17} + 236196q^{18} + 1207580q^{19} + 4209677q^{20} - 1721169q^{21} + 5460442q^{22} + 2421966q^{23} - 764235q^{24} + 7441842q^{25} - 2736526q^{26} - 11691702q^{27} + 4095246q^{28} - 2320594q^{29} - 5543721q^{30} - 3178024q^{31} - 20786874q^{32} + 5510673q^{33} - 13809336q^{34} - 2630800q^{35} + 37515798q^{36} + 3981807q^{37} - 24156377q^{38} - 22989177q^{39} - 29544450q^{40} - 885225q^{41} - 6503085q^{42} + 12360835q^{43} - 117711882q^{44} + 5301288q^{45} + 161066949q^{46} + 75901252q^{47} - 86481594q^{48} + 170907951q^{49} - 61318927q^{50} - 35388333q^{51} - 100762q^{52} - 34790192q^{53} - 19131876q^{54} + 151773316q^{55} - 417630344q^{56} - 97813980q^{57} - 432929294q^{58} + 266581942q^{59} - 340983837q^{60} - 290555332q^{61} + 158267098q^{62} + 139414689q^{63} - 131794443q^{64} - 650690086q^{65} - 442295802q^{66} + 86645184q^{67} + 62738541q^{68} - 196179246q^{69} + 429714610q^{70} - 36567631q^{71} + 61903035q^{72} + 907807228q^{73} - 171827242q^{74} - 602789202q^{75} + 1744504396q^{76} - 310688725q^{77} + 221658606q^{78} + 2508604687q^{79} + 3509441927q^{80} + 947027862q^{81} + 1759214793q^{82} + 2185672083q^{83} - 331714926q^{84} + 2868860198q^{85} + 2397001564q^{86} + 187968114q^{87} + 7683735877q^{88} + 1320145942q^{89} + 449041401q^{90} + 3894639897q^{91} + 3505964640q^{92} + 257419944q^{93} + 5406355552q^{94} + 3093659122q^{95} + 1683736794q^{96} + 3904552980q^{97} + 6137683116q^{98} - 446364513q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 18.6113 0.822512 0.411256 0.911520i \(-0.365090\pi\)
0.411256 + 0.911520i \(0.365090\pi\)
\(3\) −81.0000 −0.577350
\(4\) −165.619 −0.323475
\(5\) 1492.75 1.06813 0.534063 0.845445i \(-0.320664\pi\)
0.534063 + 0.845445i \(0.320664\pi\)
\(6\) −1507.52 −0.474877
\(7\) 11801.9 1.85785 0.928924 0.370271i \(-0.120735\pi\)
0.928924 + 0.370271i \(0.120735\pi\)
\(8\) −12611.4 −1.08857
\(9\) 6561.00 0.333333
\(10\) 27782.1 0.878546
\(11\) 54115.8 1.11444 0.557220 0.830365i \(-0.311868\pi\)
0.557220 + 0.830365i \(0.311868\pi\)
\(12\) 13415.1 0.186758
\(13\) 176081. 1.70989 0.854945 0.518718i \(-0.173591\pi\)
0.854945 + 0.518718i \(0.173591\pi\)
\(14\) 219649. 1.52810
\(15\) −120913. −0.616683
\(16\) −149917. −0.571889
\(17\) 108748. 0.315791 0.157896 0.987456i \(-0.449529\pi\)
0.157896 + 0.987456i \(0.449529\pi\)
\(18\) 122109. 0.274171
\(19\) 54464.9 0.0958795 0.0479397 0.998850i \(-0.484734\pi\)
0.0479397 + 0.998850i \(0.484734\pi\)
\(20\) −247228. −0.345512
\(21\) −955953. −1.07263
\(22\) 1.00717e6 0.916640
\(23\) −2.05205e6 −1.52902 −0.764511 0.644611i \(-0.777019\pi\)
−0.764511 + 0.644611i \(0.777019\pi\)
\(24\) 1.02152e6 0.628488
\(25\) 275183. 0.140894
\(26\) 3.27711e6 1.40640
\(27\) −531441. −0.192450
\(28\) −1.95462e6 −0.600967
\(29\) −3.15094e6 −0.827272 −0.413636 0.910442i \(-0.635741\pi\)
−0.413636 + 0.910442i \(0.635741\pi\)
\(30\) −2.25035e6 −0.507229
\(31\) 5.67861e6 1.10437 0.552185 0.833722i \(-0.313794\pi\)
0.552185 + 0.833722i \(0.313794\pi\)
\(32\) 3.66687e6 0.618188
\(33\) −4.38338e6 −0.643422
\(34\) 2.02394e6 0.259742
\(35\) 1.76173e7 1.98442
\(36\) −1.08663e6 −0.107825
\(37\) 1.48233e7 1.30028 0.650139 0.759815i \(-0.274711\pi\)
0.650139 + 0.759815i \(0.274711\pi\)
\(38\) 1.01366e6 0.0788620
\(39\) −1.42626e7 −0.987206
\(40\) −1.88257e7 −1.16273
\(41\) −2.73520e7 −1.51168 −0.755842 0.654754i \(-0.772772\pi\)
−0.755842 + 0.654754i \(0.772772\pi\)
\(42\) −1.77915e7 −0.882250
\(43\) 1.96666e7 0.877247 0.438623 0.898671i \(-0.355466\pi\)
0.438623 + 0.898671i \(0.355466\pi\)
\(44\) −8.96260e6 −0.360493
\(45\) 9.79394e6 0.356042
\(46\) −3.81914e7 −1.25764
\(47\) −1.60710e7 −0.480400 −0.240200 0.970723i \(-0.577213\pi\)
−0.240200 + 0.970723i \(0.577213\pi\)
\(48\) 1.21433e7 0.330181
\(49\) 9.89308e7 2.45160
\(50\) 5.12152e6 0.115887
\(51\) −8.80857e6 −0.182322
\(52\) −2.91624e7 −0.553106
\(53\) −4.91083e7 −0.854896 −0.427448 0.904040i \(-0.640587\pi\)
−0.427448 + 0.904040i \(0.640587\pi\)
\(54\) −9.89081e6 −0.158292
\(55\) 8.07814e7 1.19036
\(56\) −1.48838e8 −2.02240
\(57\) −4.41166e6 −0.0553561
\(58\) −5.86430e7 −0.680441
\(59\) 1.21174e7 0.130189
\(60\) 2.00255e7 0.199481
\(61\) 9.09669e7 0.841200 0.420600 0.907246i \(-0.361820\pi\)
0.420600 + 0.907246i \(0.361820\pi\)
\(62\) 1.05686e8 0.908357
\(63\) 7.74322e7 0.619283
\(64\) 1.45003e8 1.08036
\(65\) 2.62846e8 1.82638
\(66\) −8.15804e7 −0.529222
\(67\) −1.83319e7 −0.111140 −0.0555700 0.998455i \(-0.517698\pi\)
−0.0555700 + 0.998455i \(0.517698\pi\)
\(68\) −1.80107e7 −0.102150
\(69\) 1.66216e8 0.882781
\(70\) 3.27881e8 1.63221
\(71\) −3.08979e8 −1.44300 −0.721500 0.692414i \(-0.756547\pi\)
−0.721500 + 0.692414i \(0.756547\pi\)
\(72\) −8.27433e7 −0.362858
\(73\) 1.21812e7 0.0502039 0.0251019 0.999685i \(-0.492009\pi\)
0.0251019 + 0.999685i \(0.492009\pi\)
\(74\) 2.75880e8 1.06949
\(75\) −2.22898e7 −0.0813451
\(76\) −9.02043e6 −0.0310146
\(77\) 6.38668e8 2.07046
\(78\) −2.65446e8 −0.811988
\(79\) −2.20744e8 −0.637626 −0.318813 0.947818i \(-0.603284\pi\)
−0.318813 + 0.947818i \(0.603284\pi\)
\(80\) −2.23789e8 −0.610850
\(81\) 4.30467e7 0.111111
\(82\) −5.09056e8 −1.24338
\(83\) 4.48981e8 1.03843 0.519215 0.854644i \(-0.326224\pi\)
0.519215 + 0.854644i \(0.326224\pi\)
\(84\) 1.58324e8 0.346968
\(85\) 1.62333e8 0.337305
\(86\) 3.66022e8 0.721546
\(87\) 2.55226e8 0.477626
\(88\) −6.82475e8 −1.21315
\(89\) 7.18855e8 1.21447 0.607234 0.794523i \(-0.292279\pi\)
0.607234 + 0.794523i \(0.292279\pi\)
\(90\) 1.82278e8 0.292849
\(91\) 2.07809e9 3.17672
\(92\) 3.39859e8 0.494600
\(93\) −4.59968e8 −0.637608
\(94\) −2.99103e8 −0.395134
\(95\) 8.13026e7 0.102411
\(96\) −2.97016e8 −0.356911
\(97\) −1.57982e9 −1.81191 −0.905953 0.423378i \(-0.860844\pi\)
−0.905953 + 0.423378i \(0.860844\pi\)
\(98\) 1.84123e9 2.01647
\(99\) 3.55054e8 0.371480
\(100\) −4.55756e7 −0.0455756
\(101\) 4.78446e8 0.457495 0.228748 0.973486i \(-0.426537\pi\)
0.228748 + 0.973486i \(0.426537\pi\)
\(102\) −1.63939e8 −0.149962
\(103\) 1.88156e9 1.64722 0.823608 0.567160i \(-0.191958\pi\)
0.823608 + 0.567160i \(0.191958\pi\)
\(104\) −2.22063e9 −1.86134
\(105\) −1.42700e9 −1.14570
\(106\) −9.13970e8 −0.703162
\(107\) 1.55230e9 1.14485 0.572426 0.819956i \(-0.306002\pi\)
0.572426 + 0.819956i \(0.306002\pi\)
\(108\) 8.80167e7 0.0622527
\(109\) 2.26041e9 1.53380 0.766898 0.641769i \(-0.221799\pi\)
0.766898 + 0.641769i \(0.221799\pi\)
\(110\) 1.50345e9 0.979087
\(111\) −1.20068e9 −0.750716
\(112\) −1.76931e9 −1.06248
\(113\) 5.97898e8 0.344964 0.172482 0.985013i \(-0.444821\pi\)
0.172482 + 0.985013i \(0.444821\pi\)
\(114\) −8.21068e7 −0.0455310
\(115\) −3.06321e9 −1.63319
\(116\) 5.21855e8 0.267602
\(117\) 1.15527e9 0.569964
\(118\) 2.25520e8 0.107082
\(119\) 1.28343e9 0.586692
\(120\) 1.52488e9 0.671305
\(121\) 5.70570e8 0.241977
\(122\) 1.69301e9 0.691897
\(123\) 2.21551e9 0.872771
\(124\) −9.40486e8 −0.357236
\(125\) −2.50475e9 −0.917634
\(126\) 1.44111e9 0.509367
\(127\) −1.45243e9 −0.495425 −0.247712 0.968834i \(-0.579679\pi\)
−0.247712 + 0.968834i \(0.579679\pi\)
\(128\) 8.21259e8 0.270418
\(129\) −1.59300e9 −0.506479
\(130\) 4.89191e9 1.50222
\(131\) 6.58148e8 0.195255 0.0976276 0.995223i \(-0.468875\pi\)
0.0976276 + 0.995223i \(0.468875\pi\)
\(132\) 7.25971e8 0.208131
\(133\) 6.42789e8 0.178129
\(134\) −3.41180e8 −0.0914139
\(135\) −7.93310e8 −0.205561
\(136\) −1.37146e9 −0.343762
\(137\) −2.80147e9 −0.679429 −0.339714 0.940529i \(-0.610330\pi\)
−0.339714 + 0.940529i \(0.610330\pi\)
\(138\) 3.09351e9 0.726098
\(139\) −8.04977e9 −1.82901 −0.914507 0.404571i \(-0.867421\pi\)
−0.914507 + 0.404571i \(0.867421\pi\)
\(140\) −2.91776e9 −0.641908
\(141\) 1.30175e9 0.277359
\(142\) −5.75051e9 −1.18688
\(143\) 9.52878e9 1.90557
\(144\) −9.83608e8 −0.190630
\(145\) −4.70356e9 −0.883631
\(146\) 2.26708e8 0.0412933
\(147\) −8.01340e9 −1.41543
\(148\) −2.45501e9 −0.420607
\(149\) −1.17236e9 −0.194861 −0.0974304 0.995242i \(-0.531062\pi\)
−0.0974304 + 0.995242i \(0.531062\pi\)
\(150\) −4.14843e8 −0.0669072
\(151\) 1.25208e9 0.195991 0.0979954 0.995187i \(-0.468757\pi\)
0.0979954 + 0.995187i \(0.468757\pi\)
\(152\) −6.86878e8 −0.104372
\(153\) 7.13494e8 0.105264
\(154\) 1.18865e10 1.70298
\(155\) 8.47676e9 1.17961
\(156\) 2.36216e9 0.319336
\(157\) 6.13909e9 0.806410 0.403205 0.915110i \(-0.367896\pi\)
0.403205 + 0.915110i \(0.367896\pi\)
\(158\) −4.10833e9 −0.524455
\(159\) 3.97777e9 0.493575
\(160\) 5.47372e9 0.660302
\(161\) −2.42181e10 −2.84069
\(162\) 8.01156e8 0.0913902
\(163\) 1.04317e10 1.15748 0.578738 0.815513i \(-0.303545\pi\)
0.578738 + 0.815513i \(0.303545\pi\)
\(164\) 4.53000e9 0.488992
\(165\) −6.54330e9 −0.687256
\(166\) 8.35613e9 0.854120
\(167\) 6.18162e9 0.615004 0.307502 0.951547i \(-0.400507\pi\)
0.307502 + 0.951547i \(0.400507\pi\)
\(168\) 1.20559e10 1.16764
\(169\) 2.04002e10 1.92373
\(170\) 3.02124e9 0.277437
\(171\) 3.57344e8 0.0319598
\(172\) −3.25717e9 −0.283767
\(173\) −1.19474e10 −1.01407 −0.507033 0.861927i \(-0.669258\pi\)
−0.507033 + 0.861927i \(0.669258\pi\)
\(174\) 4.75009e9 0.392853
\(175\) 3.24768e9 0.261759
\(176\) −8.11290e9 −0.637337
\(177\) −9.81506e8 −0.0751646
\(178\) 1.33788e10 0.998914
\(179\) 1.17881e10 0.858230 0.429115 0.903250i \(-0.358826\pi\)
0.429115 + 0.903250i \(0.358826\pi\)
\(180\) −1.62206e9 −0.115171
\(181\) 1.59490e10 1.10453 0.552266 0.833668i \(-0.313763\pi\)
0.552266 + 0.833668i \(0.313763\pi\)
\(182\) 3.86760e10 2.61289
\(183\) −7.36832e9 −0.485667
\(184\) 2.58792e10 1.66445
\(185\) 2.21275e10 1.38886
\(186\) −8.56060e9 −0.524440
\(187\) 5.88497e9 0.351931
\(188\) 2.66166e9 0.155397
\(189\) −6.27200e9 −0.357543
\(190\) 1.51315e9 0.0842346
\(191\) −2.20091e10 −1.19661 −0.598305 0.801269i \(-0.704159\pi\)
−0.598305 + 0.801269i \(0.704159\pi\)
\(192\) −1.17452e10 −0.623744
\(193\) −2.16486e10 −1.12311 −0.561556 0.827439i \(-0.689797\pi\)
−0.561556 + 0.827439i \(0.689797\pi\)
\(194\) −2.94026e10 −1.49031
\(195\) −2.12905e10 −1.05446
\(196\) −1.63848e10 −0.793030
\(197\) 1.45366e10 0.687647 0.343823 0.939034i \(-0.388278\pi\)
0.343823 + 0.939034i \(0.388278\pi\)
\(198\) 6.60801e9 0.305547
\(199\) −2.43774e10 −1.10192 −0.550958 0.834533i \(-0.685737\pi\)
−0.550958 + 0.834533i \(0.685737\pi\)
\(200\) −3.47044e9 −0.153373
\(201\) 1.48488e9 0.0641667
\(202\) 8.90450e9 0.376295
\(203\) −3.71870e10 −1.53695
\(204\) 1.45887e9 0.0589766
\(205\) −4.08297e10 −1.61467
\(206\) 3.50183e10 1.35485
\(207\) −1.34635e10 −0.509674
\(208\) −2.63977e10 −0.977868
\(209\) 2.94741e9 0.106852
\(210\) −2.65583e10 −0.942354
\(211\) −1.54719e10 −0.537370 −0.268685 0.963228i \(-0.586589\pi\)
−0.268685 + 0.963228i \(0.586589\pi\)
\(212\) 8.13327e9 0.276537
\(213\) 2.50273e10 0.833117
\(214\) 2.88904e10 0.941655
\(215\) 2.93574e10 0.937010
\(216\) 6.70220e9 0.209496
\(217\) 6.70183e10 2.05175
\(218\) 4.20691e10 1.26156
\(219\) −9.86678e8 −0.0289852
\(220\) −1.33789e10 −0.385052
\(221\) 1.91485e10 0.539969
\(222\) −2.23463e10 −0.617472
\(223\) −2.66817e10 −0.722505 −0.361253 0.932468i \(-0.617651\pi\)
−0.361253 + 0.932468i \(0.617651\pi\)
\(224\) 4.32759e10 1.14850
\(225\) 1.80548e9 0.0469646
\(226\) 1.11277e10 0.283737
\(227\) −2.11753e10 −0.529315 −0.264657 0.964343i \(-0.585259\pi\)
−0.264657 + 0.964343i \(0.585259\pi\)
\(228\) 7.30655e8 0.0179063
\(229\) −3.86876e10 −0.929634 −0.464817 0.885407i \(-0.653880\pi\)
−0.464817 + 0.885407i \(0.653880\pi\)
\(230\) −5.70103e10 −1.34332
\(231\) −5.17321e10 −1.19538
\(232\) 3.97376e10 0.900547
\(233\) −2.74168e10 −0.609418 −0.304709 0.952446i \(-0.598559\pi\)
−0.304709 + 0.952446i \(0.598559\pi\)
\(234\) 2.15011e10 0.468802
\(235\) −2.39900e10 −0.513128
\(236\) −2.00687e9 −0.0421128
\(237\) 1.78802e10 0.368134
\(238\) 2.38863e10 0.482561
\(239\) 7.19297e10 1.42599 0.712997 0.701167i \(-0.247337\pi\)
0.712997 + 0.701167i \(0.247337\pi\)
\(240\) 1.81269e10 0.352675
\(241\) 1.36309e10 0.260283 0.130142 0.991495i \(-0.458457\pi\)
0.130142 + 0.991495i \(0.458457\pi\)
\(242\) 1.06191e10 0.199029
\(243\) −3.48678e9 −0.0641500
\(244\) −1.50658e10 −0.272107
\(245\) 1.47679e11 2.61862
\(246\) 4.12335e10 0.717864
\(247\) 9.59026e9 0.163943
\(248\) −7.16151e10 −1.20219
\(249\) −3.63675e10 −0.599538
\(250\) −4.66167e10 −0.754765
\(251\) 2.60960e10 0.414994 0.207497 0.978236i \(-0.433468\pi\)
0.207497 + 0.978236i \(0.433468\pi\)
\(252\) −1.28242e10 −0.200322
\(253\) −1.11049e11 −1.70400
\(254\) −2.70316e10 −0.407493
\(255\) −1.31490e10 −0.194743
\(256\) −5.89568e10 −0.857934
\(257\) −7.00925e10 −1.00224 −0.501121 0.865377i \(-0.667079\pi\)
−0.501121 + 0.865377i \(0.667079\pi\)
\(258\) −2.96478e10 −0.416585
\(259\) 1.74942e11 2.41572
\(260\) −4.35323e10 −0.590787
\(261\) −2.06733e10 −0.275757
\(262\) 1.22490e10 0.160600
\(263\) 1.15879e11 1.49350 0.746750 0.665105i \(-0.231613\pi\)
0.746750 + 0.665105i \(0.231613\pi\)
\(264\) 5.52804e10 0.700413
\(265\) −7.33065e10 −0.913137
\(266\) 1.19631e10 0.146514
\(267\) −5.82272e10 −0.701173
\(268\) 3.03611e9 0.0359510
\(269\) 1.49016e11 1.73519 0.867597 0.497268i \(-0.165663\pi\)
0.867597 + 0.497268i \(0.165663\pi\)
\(270\) −1.47645e10 −0.169076
\(271\) −1.47037e11 −1.65602 −0.828009 0.560715i \(-0.810526\pi\)
−0.828009 + 0.560715i \(0.810526\pi\)
\(272\) −1.63032e10 −0.180598
\(273\) −1.68325e11 −1.83408
\(274\) −5.21391e10 −0.558838
\(275\) 1.48918e10 0.157018
\(276\) −2.75286e10 −0.285557
\(277\) 1.15628e11 1.18006 0.590032 0.807380i \(-0.299115\pi\)
0.590032 + 0.807380i \(0.299115\pi\)
\(278\) −1.49817e11 −1.50438
\(279\) 3.72574e10 0.368123
\(280\) −2.22178e11 −2.16018
\(281\) −1.10467e11 −1.05695 −0.528476 0.848948i \(-0.677236\pi\)
−0.528476 + 0.848948i \(0.677236\pi\)
\(282\) 2.42273e10 0.228131
\(283\) −1.39797e11 −1.29557 −0.647783 0.761825i \(-0.724303\pi\)
−0.647783 + 0.761825i \(0.724303\pi\)
\(284\) 5.11728e10 0.466774
\(285\) −6.58551e9 −0.0591273
\(286\) 1.77343e11 1.56735
\(287\) −3.22805e11 −2.80848
\(288\) 2.40583e10 0.206063
\(289\) −1.06762e11 −0.900276
\(290\) −8.75395e10 −0.726797
\(291\) 1.27966e11 1.04610
\(292\) −2.01744e9 −0.0162397
\(293\) −2.95432e10 −0.234182 −0.117091 0.993121i \(-0.537357\pi\)
−0.117091 + 0.993121i \(0.537357\pi\)
\(294\) −1.49140e11 −1.16421
\(295\) 1.80882e10 0.139058
\(296\) −1.86942e11 −1.41545
\(297\) −2.87593e10 −0.214474
\(298\) −2.18192e10 −0.160275
\(299\) −3.61329e11 −2.61446
\(300\) 3.69162e9 0.0263131
\(301\) 2.32103e11 1.62979
\(302\) 2.33029e10 0.161205
\(303\) −3.87541e10 −0.264135
\(304\) −8.16524e9 −0.0548325
\(305\) 1.35791e11 0.898508
\(306\) 1.32791e10 0.0865807
\(307\) −5.96724e9 −0.0383399 −0.0191700 0.999816i \(-0.506102\pi\)
−0.0191700 + 0.999816i \(0.506102\pi\)
\(308\) −1.05776e11 −0.669742
\(309\) −1.52406e11 −0.951021
\(310\) 1.57764e11 0.970240
\(311\) −2.69967e11 −1.63640 −0.818199 0.574934i \(-0.805028\pi\)
−0.818199 + 0.574934i \(0.805028\pi\)
\(312\) 1.79871e11 1.07465
\(313\) −3.88607e10 −0.228855 −0.114428 0.993432i \(-0.536503\pi\)
−0.114428 + 0.993432i \(0.536503\pi\)
\(314\) 1.14257e11 0.663281
\(315\) 1.15587e11 0.661472
\(316\) 3.65593e10 0.206256
\(317\) −2.23628e11 −1.24383 −0.621913 0.783086i \(-0.713644\pi\)
−0.621913 + 0.783086i \(0.713644\pi\)
\(318\) 7.40316e10 0.405971
\(319\) −1.70515e11 −0.921946
\(320\) 2.16453e11 1.15396
\(321\) −1.25737e11 −0.660981
\(322\) −4.50731e11 −2.33650
\(323\) 5.92294e9 0.0302779
\(324\) −7.12936e9 −0.0359416
\(325\) 4.84546e10 0.240913
\(326\) 1.94148e11 0.952038
\(327\) −1.83093e11 −0.885537
\(328\) 3.44946e11 1.64558
\(329\) −1.89668e11 −0.892510
\(330\) −1.21779e11 −0.565276
\(331\) 3.56601e11 1.63289 0.816444 0.577425i \(-0.195942\pi\)
0.816444 + 0.577425i \(0.195942\pi\)
\(332\) −7.43599e10 −0.335906
\(333\) 9.72554e10 0.433426
\(334\) 1.15048e11 0.505848
\(335\) −2.73649e10 −0.118712
\(336\) 1.43314e11 0.613425
\(337\) 4.09649e11 1.73012 0.865061 0.501666i \(-0.167279\pi\)
0.865061 + 0.501666i \(0.167279\pi\)
\(338\) 3.79674e11 1.58229
\(339\) −4.84298e10 −0.199165
\(340\) −2.68855e10 −0.109110
\(341\) 3.07303e11 1.23075
\(342\) 6.65065e9 0.0262873
\(343\) 6.91322e11 2.69685
\(344\) −2.48023e11 −0.954947
\(345\) 2.48120e11 0.942922
\(346\) −2.22357e11 −0.834081
\(347\) 2.00285e11 0.741595 0.370797 0.928714i \(-0.379084\pi\)
0.370797 + 0.928714i \(0.379084\pi\)
\(348\) −4.22702e10 −0.154500
\(349\) 4.58740e11 1.65521 0.827603 0.561314i \(-0.189704\pi\)
0.827603 + 0.561314i \(0.189704\pi\)
\(350\) 6.04436e10 0.215300
\(351\) −9.35769e10 −0.329069
\(352\) 1.98435e11 0.688933
\(353\) 2.90410e11 0.995463 0.497732 0.867331i \(-0.334166\pi\)
0.497732 + 0.867331i \(0.334166\pi\)
\(354\) −1.82671e10 −0.0618238
\(355\) −4.61229e11 −1.54131
\(356\) −1.19056e11 −0.392850
\(357\) −1.03958e11 −0.338727
\(358\) 2.19391e11 0.705904
\(359\) 8.27981e10 0.263084 0.131542 0.991311i \(-0.458007\pi\)
0.131542 + 0.991311i \(0.458007\pi\)
\(360\) −1.23515e11 −0.387578
\(361\) −3.19721e11 −0.990807
\(362\) 2.96831e11 0.908491
\(363\) −4.62162e10 −0.139706
\(364\) −3.44172e11 −1.02759
\(365\) 1.81835e10 0.0536241
\(366\) −1.37134e11 −0.399467
\(367\) 3.34608e11 0.962808 0.481404 0.876499i \(-0.340127\pi\)
0.481404 + 0.876499i \(0.340127\pi\)
\(368\) 3.07639e11 0.874431
\(369\) −1.79456e11 −0.503895
\(370\) 4.11821e11 1.14235
\(371\) −5.79570e11 −1.58827
\(372\) 7.61794e10 0.206250
\(373\) 4.59899e11 1.23019 0.615096 0.788453i \(-0.289117\pi\)
0.615096 + 0.788453i \(0.289117\pi\)
\(374\) 1.09527e11 0.289467
\(375\) 2.02885e11 0.529796
\(376\) 2.02678e11 0.522950
\(377\) −5.54821e11 −1.41455
\(378\) −1.16730e11 −0.294083
\(379\) −4.64799e10 −0.115715 −0.0578574 0.998325i \(-0.518427\pi\)
−0.0578574 + 0.998325i \(0.518427\pi\)
\(380\) −1.34653e10 −0.0331275
\(381\) 1.17647e11 0.286034
\(382\) −4.09618e11 −0.984225
\(383\) −3.29063e11 −0.781419 −0.390710 0.920514i \(-0.627770\pi\)
−0.390710 + 0.920514i \(0.627770\pi\)
\(384\) −6.65220e10 −0.156126
\(385\) 9.53373e11 2.21151
\(386\) −4.02910e11 −0.923772
\(387\) 1.29033e11 0.292416
\(388\) 2.61649e11 0.586106
\(389\) −1.15550e11 −0.255857 −0.127929 0.991783i \(-0.540833\pi\)
−0.127929 + 0.991783i \(0.540833\pi\)
\(390\) −3.96244e11 −0.867306
\(391\) −2.23156e11 −0.482852
\(392\) −1.24765e12 −2.66874
\(393\) −5.33100e10 −0.112731
\(394\) 2.70546e11 0.565598
\(395\) −3.29515e11 −0.681065
\(396\) −5.88036e10 −0.120164
\(397\) −3.49147e11 −0.705425 −0.352713 0.935732i \(-0.614741\pi\)
−0.352713 + 0.935732i \(0.614741\pi\)
\(398\) −4.53695e11 −0.906338
\(399\) −5.20659e10 −0.102843
\(400\) −4.12547e10 −0.0805757
\(401\) 5.89365e11 1.13824 0.569122 0.822253i \(-0.307283\pi\)
0.569122 + 0.822253i \(0.307283\pi\)
\(402\) 2.76356e10 0.0527778
\(403\) 9.99898e11 1.88835
\(404\) −7.92397e10 −0.147988
\(405\) 6.42581e10 0.118681
\(406\) −6.92098e11 −1.26416
\(407\) 8.02173e11 1.44908
\(408\) 1.11088e11 0.198471
\(409\) 3.27083e11 0.577967 0.288983 0.957334i \(-0.406683\pi\)
0.288983 + 0.957334i \(0.406683\pi\)
\(410\) −7.59894e11 −1.32808
\(411\) 2.26919e11 0.392268
\(412\) −3.11622e11 −0.532833
\(413\) 1.43008e11 0.241871
\(414\) −2.50574e11 −0.419213
\(415\) 6.70218e11 1.10917
\(416\) 6.45667e11 1.05703
\(417\) 6.52031e11 1.05598
\(418\) 5.48552e10 0.0878870
\(419\) 4.85479e11 0.769497 0.384749 0.923021i \(-0.374288\pi\)
0.384749 + 0.923021i \(0.374288\pi\)
\(420\) 2.36338e11 0.370606
\(421\) −6.01168e9 −0.00932667 −0.00466333 0.999989i \(-0.501484\pi\)
−0.00466333 + 0.999989i \(0.501484\pi\)
\(422\) −2.87953e11 −0.441993
\(423\) −1.05442e11 −0.160133
\(424\) 6.19323e11 0.930617
\(425\) 2.99255e10 0.0444930
\(426\) 4.65791e11 0.685248
\(427\) 1.07358e12 1.56282
\(428\) −2.57091e11 −0.370331
\(429\) −7.71831e11 −1.10018
\(430\) 5.46379e11 0.770702
\(431\) 2.61315e11 0.364768 0.182384 0.983227i \(-0.441619\pi\)
0.182384 + 0.983227i \(0.441619\pi\)
\(432\) 7.96722e10 0.110060
\(433\) −1.12179e12 −1.53361 −0.766805 0.641881i \(-0.778154\pi\)
−0.766805 + 0.641881i \(0.778154\pi\)
\(434\) 1.24730e12 1.68759
\(435\) 3.80989e11 0.510165
\(436\) −3.74366e11 −0.496144
\(437\) −1.11765e11 −0.146602
\(438\) −1.83634e10 −0.0238407
\(439\) 2.31501e11 0.297483 0.148742 0.988876i \(-0.452478\pi\)
0.148742 + 0.988876i \(0.452478\pi\)
\(440\) −1.01877e12 −1.29580
\(441\) 6.49085e11 0.817199
\(442\) 3.56378e11 0.444130
\(443\) 2.00771e11 0.247676 0.123838 0.992302i \(-0.460480\pi\)
0.123838 + 0.992302i \(0.460480\pi\)
\(444\) 1.98856e11 0.242837
\(445\) 1.07307e12 1.29721
\(446\) −4.96581e11 −0.594269
\(447\) 9.49615e10 0.112503
\(448\) 1.71131e12 2.00714
\(449\) −3.68935e11 −0.428392 −0.214196 0.976791i \(-0.568713\pi\)
−0.214196 + 0.976791i \(0.568713\pi\)
\(450\) 3.36023e10 0.0386289
\(451\) −1.48017e12 −1.68468
\(452\) −9.90233e10 −0.111587
\(453\) −1.01418e11 −0.113155
\(454\) −3.94101e11 −0.435368
\(455\) 3.10208e12 3.39313
\(456\) 5.56371e10 0.0602591
\(457\) −1.37268e12 −1.47213 −0.736066 0.676910i \(-0.763319\pi\)
−0.736066 + 0.676910i \(0.763319\pi\)
\(458\) −7.20027e11 −0.764634
\(459\) −5.77930e10 −0.0607741
\(460\) 5.07326e11 0.528295
\(461\) 4.80655e11 0.495654 0.247827 0.968804i \(-0.420283\pi\)
0.247827 + 0.968804i \(0.420283\pi\)
\(462\) −9.62803e11 −0.983215
\(463\) 2.39475e11 0.242185 0.121092 0.992641i \(-0.461360\pi\)
0.121092 + 0.992641i \(0.461360\pi\)
\(464\) 4.72380e11 0.473108
\(465\) −6.86617e11 −0.681046
\(466\) −5.10263e11 −0.501253
\(467\) −1.22103e12 −1.18796 −0.593980 0.804480i \(-0.702444\pi\)
−0.593980 + 0.804480i \(0.702444\pi\)
\(468\) −1.91335e11 −0.184369
\(469\) −2.16351e11 −0.206481
\(470\) −4.46486e11 −0.422053
\(471\) −4.97266e11 −0.465581
\(472\) −1.52817e11 −0.141720
\(473\) 1.06427e12 0.977639
\(474\) 3.32775e11 0.302794
\(475\) 1.49878e10 0.0135088
\(476\) −2.12560e11 −0.189780
\(477\) −3.22199e11 −0.284965
\(478\) 1.33871e12 1.17290
\(479\) 1.73521e12 1.50606 0.753028 0.657988i \(-0.228592\pi\)
0.753028 + 0.657988i \(0.228592\pi\)
\(480\) −4.43372e11 −0.381226
\(481\) 2.61010e12 2.22333
\(482\) 2.53688e11 0.214086
\(483\) 1.96167e12 1.64007
\(484\) −9.44973e10 −0.0782736
\(485\) −2.35828e12 −1.93534
\(486\) −6.48936e10 −0.0527641
\(487\) −1.80440e12 −1.45363 −0.726815 0.686834i \(-0.759000\pi\)
−0.726815 + 0.686834i \(0.759000\pi\)
\(488\) −1.14722e12 −0.915708
\(489\) −8.44970e11 −0.668269
\(490\) 2.74850e12 2.15384
\(491\) −1.81935e12 −1.41270 −0.706351 0.707862i \(-0.749660\pi\)
−0.706351 + 0.707862i \(0.749660\pi\)
\(492\) −3.66930e11 −0.282319
\(493\) −3.42657e11 −0.261245
\(494\) 1.78487e11 0.134845
\(495\) 5.30007e11 0.396788
\(496\) −8.51323e11 −0.631578
\(497\) −3.64653e12 −2.68088
\(498\) −6.76847e11 −0.493127
\(499\) −3.19828e11 −0.230921 −0.115461 0.993312i \(-0.536834\pi\)
−0.115461 + 0.993312i \(0.536834\pi\)
\(500\) 4.14834e11 0.296831
\(501\) −5.00711e11 −0.355073
\(502\) 4.85680e11 0.341337
\(503\) 8.65153e11 0.602611 0.301305 0.953528i \(-0.402578\pi\)
0.301305 + 0.953528i \(0.402578\pi\)
\(504\) −9.76526e11 −0.674134
\(505\) 7.14201e11 0.488663
\(506\) −2.06676e12 −1.40156
\(507\) −1.65241e12 −1.11066
\(508\) 2.40550e11 0.160257
\(509\) 1.81057e12 1.19560 0.597798 0.801647i \(-0.296042\pi\)
0.597798 + 0.801647i \(0.296042\pi\)
\(510\) −2.44720e11 −0.160178
\(511\) 1.43761e11 0.0932712
\(512\) −1.51775e12 −0.976079
\(513\) −2.89449e10 −0.0184520
\(514\) −1.30451e12 −0.824356
\(515\) 2.80870e12 1.75943
\(516\) 2.63831e11 0.163833
\(517\) −8.69695e11 −0.535377
\(518\) 3.25591e12 1.98696
\(519\) 9.67740e11 0.585471
\(520\) −3.31485e12 −1.98815
\(521\) −2.25238e12 −1.33928 −0.669640 0.742686i \(-0.733552\pi\)
−0.669640 + 0.742686i \(0.733552\pi\)
\(522\) −3.84757e11 −0.226814
\(523\) −3.18471e11 −0.186128 −0.0930640 0.995660i \(-0.529666\pi\)
−0.0930640 + 0.995660i \(0.529666\pi\)
\(524\) −1.09002e11 −0.0631601
\(525\) −2.63062e11 −0.151127
\(526\) 2.15667e12 1.22842
\(527\) 6.17536e11 0.348750
\(528\) 6.57145e11 0.367967
\(529\) 2.40978e12 1.33791
\(530\) −1.36433e12 −0.751066
\(531\) 7.95020e10 0.0433963
\(532\) −1.06458e11 −0.0576204
\(533\) −4.81617e12 −2.58481
\(534\) −1.08369e12 −0.576723
\(535\) 2.31720e12 1.22285
\(536\) 2.31190e11 0.120984
\(537\) −9.54833e11 −0.495499
\(538\) 2.77339e12 1.42722
\(539\) 5.35372e12 2.73216
\(540\) 1.31387e11 0.0664938
\(541\) −1.83667e11 −0.0921816 −0.0460908 0.998937i \(-0.514676\pi\)
−0.0460908 + 0.998937i \(0.514676\pi\)
\(542\) −2.73655e12 −1.36209
\(543\) −1.29187e12 −0.637702
\(544\) 3.98764e11 0.195218
\(545\) 3.37423e12 1.63829
\(546\) −3.13276e12 −1.50855
\(547\) 1.87270e12 0.894386 0.447193 0.894437i \(-0.352424\pi\)
0.447193 + 0.894437i \(0.352424\pi\)
\(548\) 4.63977e11 0.219778
\(549\) 5.96834e11 0.280400
\(550\) 2.77155e11 0.129149
\(551\) −1.71615e11 −0.0793185
\(552\) −2.09622e12 −0.960972
\(553\) −2.60519e12 −1.18461
\(554\) 2.15200e12 0.970617
\(555\) −1.79232e12 −0.801859
\(556\) 1.33320e12 0.591640
\(557\) −3.46768e12 −1.52648 −0.763239 0.646116i \(-0.776392\pi\)
−0.763239 + 0.646116i \(0.776392\pi\)
\(558\) 6.93409e11 0.302786
\(559\) 3.46293e12 1.50000
\(560\) −2.64114e12 −1.13487
\(561\) −4.76683e11 −0.203187
\(562\) −2.05594e12 −0.869355
\(563\) −1.63273e12 −0.684898 −0.342449 0.939536i \(-0.611256\pi\)
−0.342449 + 0.939536i \(0.611256\pi\)
\(564\) −2.15595e11 −0.0897186
\(565\) 8.92514e11 0.368466
\(566\) −2.60181e12 −1.06562
\(567\) 5.08032e11 0.206428
\(568\) 3.89665e12 1.57081
\(569\) −2.02549e12 −0.810075 −0.405038 0.914300i \(-0.632742\pi\)
−0.405038 + 0.914300i \(0.632742\pi\)
\(570\) −1.22565e11 −0.0486329
\(571\) 1.28759e12 0.506893 0.253447 0.967349i \(-0.418436\pi\)
0.253447 + 0.967349i \(0.418436\pi\)
\(572\) −1.57815e12 −0.616404
\(573\) 1.78274e12 0.690863
\(574\) −6.00782e12 −2.31001
\(575\) −5.64691e11 −0.215430
\(576\) 9.51364e11 0.360119
\(577\) −4.89310e12 −1.83778 −0.918889 0.394515i \(-0.870913\pi\)
−0.918889 + 0.394515i \(0.870913\pi\)
\(578\) −1.98698e12 −0.740487
\(579\) 1.75354e12 0.648429
\(580\) 7.79000e11 0.285832
\(581\) 5.29883e12 1.92924
\(582\) 2.38161e12 0.860433
\(583\) −2.65753e12 −0.952731
\(584\) −1.53622e11 −0.0546506
\(585\) 1.72453e12 0.608793
\(586\) −5.49838e11 −0.192617
\(587\) −1.75813e12 −0.611195 −0.305598 0.952161i \(-0.598856\pi\)
−0.305598 + 0.952161i \(0.598856\pi\)
\(588\) 1.32717e12 0.457856
\(589\) 3.09285e11 0.105886
\(590\) 3.36645e11 0.114377
\(591\) −1.17747e12 −0.397013
\(592\) −2.22226e12 −0.743615
\(593\) −1.62832e12 −0.540745 −0.270372 0.962756i \(-0.587147\pi\)
−0.270372 + 0.962756i \(0.587147\pi\)
\(594\) −5.35249e11 −0.176407
\(595\) 1.91584e12 0.626661
\(596\) 1.94166e11 0.0630325
\(597\) 1.97457e12 0.636191
\(598\) −6.72480e12 −2.15042
\(599\) 2.05355e12 0.651754 0.325877 0.945412i \(-0.394340\pi\)
0.325877 + 0.945412i \(0.394340\pi\)
\(600\) 2.81106e11 0.0885500
\(601\) −4.87509e12 −1.52422 −0.762110 0.647448i \(-0.775836\pi\)
−0.762110 + 0.647448i \(0.775836\pi\)
\(602\) 4.31974e12 1.34052
\(603\) −1.20275e11 −0.0370467
\(604\) −2.07368e11 −0.0633981
\(605\) 8.51720e11 0.258462
\(606\) −7.21265e11 −0.217254
\(607\) −3.47527e12 −1.03906 −0.519528 0.854453i \(-0.673892\pi\)
−0.519528 + 0.854453i \(0.673892\pi\)
\(608\) 1.99716e11 0.0592715
\(609\) 3.01214e12 0.887356
\(610\) 2.52725e12 0.739033
\(611\) −2.82981e12 −0.821431
\(612\) −1.18168e11 −0.0340502
\(613\) 2.62214e12 0.750038 0.375019 0.927017i \(-0.377636\pi\)
0.375019 + 0.927017i \(0.377636\pi\)
\(614\) −1.11058e11 −0.0315350
\(615\) 3.30720e12 0.932230
\(616\) −8.05449e12 −2.25385
\(617\) −3.35752e12 −0.932685 −0.466342 0.884604i \(-0.654429\pi\)
−0.466342 + 0.884604i \(0.654429\pi\)
\(618\) −2.83648e12 −0.782225
\(619\) 3.47281e12 0.950766 0.475383 0.879779i \(-0.342309\pi\)
0.475383 + 0.879779i \(0.342309\pi\)
\(620\) −1.40391e12 −0.381573
\(621\) 1.09055e12 0.294260
\(622\) −5.02444e12 −1.34596
\(623\) 8.48384e12 2.25630
\(624\) 2.13821e12 0.564573
\(625\) −4.27644e12 −1.12104
\(626\) −7.23249e11 −0.188236
\(627\) −2.38740e11 −0.0616910
\(628\) −1.01675e12 −0.260853
\(629\) 1.61200e12 0.410616
\(630\) 2.15123e12 0.544068
\(631\) −2.45725e12 −0.617045 −0.308523 0.951217i \(-0.599835\pi\)
−0.308523 + 0.951217i \(0.599835\pi\)
\(632\) 2.78388e12 0.694103
\(633\) 1.25323e12 0.310251
\(634\) −4.16201e12 −1.02306
\(635\) −2.16811e12 −0.529176
\(636\) −6.58795e11 −0.159659
\(637\) 1.74199e13 4.19197
\(638\) −3.17351e12 −0.758311
\(639\) −2.02721e12 −0.481000
\(640\) 1.22594e12 0.288840
\(641\) 5.98365e12 1.39992 0.699962 0.714180i \(-0.253200\pi\)
0.699962 + 0.714180i \(0.253200\pi\)
\(642\) −2.34012e12 −0.543665
\(643\) 4.95464e12 1.14304 0.571522 0.820587i \(-0.306353\pi\)
0.571522 + 0.820587i \(0.306353\pi\)
\(644\) 4.01098e12 0.918891
\(645\) −2.37795e12 −0.540983
\(646\) 1.10234e11 0.0249039
\(647\) −7.46826e12 −1.67552 −0.837761 0.546037i \(-0.816136\pi\)
−0.837761 + 0.546037i \(0.816136\pi\)
\(648\) −5.42879e11 −0.120953
\(649\) 6.55740e11 0.145088
\(650\) 9.01804e11 0.198154
\(651\) −5.42848e12 −1.18458
\(652\) −1.72769e12 −0.374414
\(653\) −5.31545e12 −1.14401 −0.572007 0.820249i \(-0.693835\pi\)
−0.572007 + 0.820249i \(0.693835\pi\)
\(654\) −3.40760e12 −0.728365
\(655\) 9.82452e11 0.208557
\(656\) 4.10053e12 0.864516
\(657\) 7.99209e10 0.0167346
\(658\) −3.52997e12 −0.734099
\(659\) 3.34203e12 0.690282 0.345141 0.938551i \(-0.387831\pi\)
0.345141 + 0.938551i \(0.387831\pi\)
\(660\) 1.08369e12 0.222310
\(661\) −1.03490e12 −0.210859 −0.105430 0.994427i \(-0.533622\pi\)
−0.105430 + 0.994427i \(0.533622\pi\)
\(662\) 6.63681e12 1.34307
\(663\) −1.55102e12 −0.311751
\(664\) −5.66228e12 −1.13041
\(665\) 9.59524e11 0.190265
\(666\) 1.81005e12 0.356498
\(667\) 6.46589e12 1.26492
\(668\) −1.02379e12 −0.198938
\(669\) 2.16121e12 0.417138
\(670\) −5.09297e11 −0.0976416
\(671\) 4.92274e12 0.937467
\(672\) −3.50535e12 −0.663086
\(673\) 2.81027e12 0.528056 0.264028 0.964515i \(-0.414949\pi\)
0.264028 + 0.964515i \(0.414949\pi\)
\(674\) 7.62410e12 1.42305
\(675\) −1.46244e11 −0.0271150
\(676\) −3.37865e12 −0.622277
\(677\) 3.12734e12 0.572172 0.286086 0.958204i \(-0.407646\pi\)
0.286086 + 0.958204i \(0.407646\pi\)
\(678\) −9.01341e11 −0.163816
\(679\) −1.86449e13 −3.36625
\(680\) −2.04725e12 −0.367181
\(681\) 1.71520e12 0.305600
\(682\) 5.71930e12 1.01231
\(683\) 8.96063e12 1.57560 0.787799 0.615933i \(-0.211221\pi\)
0.787799 + 0.615933i \(0.211221\pi\)
\(684\) −5.91830e10 −0.0103382
\(685\) −4.18190e12 −0.725715
\(686\) 1.28664e13 2.21819
\(687\) 3.13369e12 0.536724
\(688\) −2.94837e12 −0.501688
\(689\) −8.64706e12 −1.46178
\(690\) 4.61784e12 0.775564
\(691\) 3.78288e11 0.0631206 0.0315603 0.999502i \(-0.489952\pi\)
0.0315603 + 0.999502i \(0.489952\pi\)
\(692\) 1.97872e12 0.328025
\(693\) 4.19030e12 0.690154
\(694\) 3.72757e12 0.609970
\(695\) −1.20163e13 −1.95362
\(696\) −3.21875e12 −0.519931
\(697\) −2.97446e12 −0.477377
\(698\) 8.53775e12 1.36143
\(699\) 2.22076e12 0.351848
\(700\) −5.37877e11 −0.0846725
\(701\) −5.57092e12 −0.871356 −0.435678 0.900103i \(-0.643491\pi\)
−0.435678 + 0.900103i \(0.643491\pi\)
\(702\) −1.74159e12 −0.270663
\(703\) 8.07348e11 0.124670
\(704\) 7.84695e12 1.20399
\(705\) 1.94319e12 0.296254
\(706\) 5.40491e12 0.818780
\(707\) 5.64656e12 0.849956
\(708\) 1.62556e11 0.0243138
\(709\) 4.49290e12 0.667757 0.333879 0.942616i \(-0.391642\pi\)
0.333879 + 0.942616i \(0.391642\pi\)
\(710\) −8.58408e12 −1.26774
\(711\) −1.44830e12 −0.212542
\(712\) −9.06575e12 −1.32204
\(713\) −1.16528e13 −1.68861
\(714\) −1.93479e12 −0.278607
\(715\) 1.42241e13 2.03539
\(716\) −1.95233e12 −0.277616
\(717\) −5.82631e12 −0.823298
\(718\) 1.54098e12 0.216390
\(719\) −3.51348e12 −0.490295 −0.245148 0.969486i \(-0.578836\pi\)
−0.245148 + 0.969486i \(0.578836\pi\)
\(720\) −1.46828e12 −0.203617
\(721\) 2.22059e13 3.06028
\(722\) −5.95043e12 −0.814950
\(723\) −1.10410e12 −0.150275
\(724\) −2.64145e12 −0.357288
\(725\) −8.67084e11 −0.116558
\(726\) −8.60144e11 −0.114910
\(727\) 2.85192e12 0.378646 0.189323 0.981915i \(-0.439371\pi\)
0.189323 + 0.981915i \(0.439371\pi\)
\(728\) −2.62076e13 −3.45809
\(729\) 2.82430e11 0.0370370
\(730\) 3.38419e11 0.0441064
\(731\) 2.13870e12 0.277027
\(732\) 1.22033e12 0.157101
\(733\) −6.61645e12 −0.846559 −0.423279 0.905999i \(-0.639121\pi\)
−0.423279 + 0.905999i \(0.639121\pi\)
\(734\) 6.22750e12 0.791921
\(735\) −1.19620e13 −1.51186
\(736\) −7.52461e12 −0.945222
\(737\) −9.92044e11 −0.123859
\(738\) −3.33992e12 −0.414459
\(739\) 6.98786e12 0.861875 0.430937 0.902382i \(-0.358183\pi\)
0.430937 + 0.902382i \(0.358183\pi\)
\(740\) −3.66473e12 −0.449261
\(741\) −7.76811e11 −0.0946528
\(742\) −1.07866e13 −1.30637
\(743\) −1.02827e13 −1.23783 −0.618913 0.785460i \(-0.712427\pi\)
−0.618913 + 0.785460i \(0.712427\pi\)
\(744\) 5.80083e12 0.694083
\(745\) −1.75005e12 −0.208136
\(746\) 8.55932e12 1.01185
\(747\) 2.94577e12 0.346143
\(748\) −9.74663e11 −0.113841
\(749\) 1.83201e13 2.12696
\(750\) 3.77595e12 0.435764
\(751\) 6.20048e12 0.711289 0.355644 0.934621i \(-0.384261\pi\)
0.355644 + 0.934621i \(0.384261\pi\)
\(752\) 2.40932e12 0.274736
\(753\) −2.11377e12 −0.239597
\(754\) −1.03259e13 −1.16348
\(755\) 1.86904e12 0.209343
\(756\) 1.03876e12 0.115656
\(757\) −8.98211e10 −0.00994139 −0.00497069 0.999988i \(-0.501582\pi\)
−0.00497069 + 0.999988i \(0.501582\pi\)
\(758\) −8.65052e11 −0.0951767
\(759\) 8.99493e12 0.983807
\(760\) −1.02534e12 −0.111482
\(761\) −8.53129e12 −0.922112 −0.461056 0.887371i \(-0.652529\pi\)
−0.461056 + 0.887371i \(0.652529\pi\)
\(762\) 2.18956e12 0.235266
\(763\) 2.66771e13 2.84956
\(764\) 3.64513e12 0.387073
\(765\) 1.06507e12 0.112435
\(766\) −6.12429e12 −0.642726
\(767\) 2.13364e12 0.222609
\(768\) 4.77550e12 0.495329
\(769\) −1.33335e13 −1.37492 −0.687458 0.726224i \(-0.741274\pi\)
−0.687458 + 0.726224i \(0.741274\pi\)
\(770\) 1.77435e13 1.81900
\(771\) 5.67750e12 0.578645
\(772\) 3.58543e12 0.363298
\(773\) −9.54484e12 −0.961525 −0.480763 0.876851i \(-0.659640\pi\)
−0.480763 + 0.876851i \(0.659640\pi\)
\(774\) 2.40147e12 0.240515
\(775\) 1.56266e12 0.155599
\(776\) 1.99238e13 1.97239
\(777\) −1.41703e13 −1.39472
\(778\) −2.15054e12 −0.210445
\(779\) −1.48972e12 −0.144940
\(780\) 3.52611e12 0.341091
\(781\) −1.67206e13 −1.60814
\(782\) −4.15323e12 −0.397151
\(783\) 1.67454e12 0.159209
\(784\) −1.48315e13 −1.40204
\(785\) 9.16414e12 0.861347
\(786\) −9.92169e11 −0.0927223
\(787\) 7.44479e12 0.691777 0.345888 0.938276i \(-0.387578\pi\)
0.345888 + 0.938276i \(0.387578\pi\)
\(788\) −2.40754e12 −0.222436
\(789\) −9.38622e12 −0.862272
\(790\) −6.13271e12 −0.560184
\(791\) 7.05633e12 0.640891
\(792\) −4.47772e12 −0.404383
\(793\) 1.60176e13 1.43836
\(794\) −6.49808e12 −0.580220
\(795\) 5.93783e12 0.527200
\(796\) 4.03736e12 0.356442
\(797\) 1.74579e13 1.53260 0.766301 0.642482i \(-0.222095\pi\)
0.766301 + 0.642482i \(0.222095\pi\)
\(798\) −9.69014e11 −0.0845897
\(799\) −1.74769e12 −0.151706
\(800\) 1.00906e12 0.0870988
\(801\) 4.71641e12 0.404823
\(802\) 1.09689e13 0.936218
\(803\) 6.59195e11 0.0559492
\(804\) −2.45925e11 −0.0207563
\(805\) −3.61516e13 −3.03422
\(806\) 1.86094e13 1.55319
\(807\) −1.20703e13 −1.00181
\(808\) −6.03386e12 −0.498017
\(809\) 7.20004e12 0.590971 0.295486 0.955347i \(-0.404519\pi\)
0.295486 + 0.955347i \(0.404519\pi\)
\(810\) 1.19593e12 0.0976163
\(811\) −1.08968e13 −0.884517 −0.442259 0.896888i \(-0.645823\pi\)
−0.442259 + 0.896888i \(0.645823\pi\)
\(812\) 6.15887e12 0.497163
\(813\) 1.19100e13 0.956102
\(814\) 1.49295e13 1.19189
\(815\) 1.55720e13 1.23633
\(816\) 1.32056e12 0.104268
\(817\) 1.07114e12 0.0841100
\(818\) 6.08744e12 0.475384
\(819\) 1.36344e13 1.05891
\(820\) 6.76217e12 0.522305
\(821\) −2.50890e13 −1.92726 −0.963629 0.267242i \(-0.913888\pi\)
−0.963629 + 0.267242i \(0.913888\pi\)
\(822\) 4.22327e12 0.322645
\(823\) −1.83804e13 −1.39655 −0.698274 0.715831i \(-0.746048\pi\)
−0.698274 + 0.715831i \(0.746048\pi\)
\(824\) −2.37291e13 −1.79312
\(825\) −1.20623e12 −0.0906542
\(826\) 2.66156e12 0.198942
\(827\) −5.81755e12 −0.432480 −0.216240 0.976340i \(-0.569379\pi\)
−0.216240 + 0.976340i \(0.569379\pi\)
\(828\) 2.22982e12 0.164867
\(829\) −7.71834e12 −0.567582 −0.283791 0.958886i \(-0.591592\pi\)
−0.283791 + 0.958886i \(0.591592\pi\)
\(830\) 1.24736e13 0.912308
\(831\) −9.36590e12 −0.681311
\(832\) 2.55323e13 1.84729
\(833\) 1.07585e13 0.774193
\(834\) 1.21352e13 0.868557
\(835\) 9.22762e12 0.656902
\(836\) −4.88148e11 −0.0345639
\(837\) −3.01785e12 −0.212536
\(838\) 9.03540e12 0.632920
\(839\) 9.00347e12 0.627308 0.313654 0.949537i \(-0.398447\pi\)
0.313654 + 0.949537i \(0.398447\pi\)
\(840\) 1.79964e13 1.24718
\(841\) −4.57875e12 −0.315620
\(842\) −1.11885e11 −0.00767129
\(843\) 8.94785e12 0.610232
\(844\) 2.56245e12 0.173826
\(845\) 3.04524e13 2.05478
\(846\) −1.96241e12 −0.131711
\(847\) 6.73380e12 0.449557
\(848\) 7.36219e12 0.488906
\(849\) 1.13236e13 0.747995
\(850\) 5.56954e11 0.0365960
\(851\) −3.04181e13 −1.98815
\(852\) −4.14500e12 −0.269492
\(853\) −3.36589e11 −0.0217685 −0.0108843 0.999941i \(-0.503465\pi\)
−0.0108843 + 0.999941i \(0.503465\pi\)
\(854\) 1.99807e13 1.28544
\(855\) 5.33427e11 0.0341371
\(856\) −1.95767e13 −1.24626
\(857\) −2.49442e12 −0.157963 −0.0789816 0.996876i \(-0.525167\pi\)
−0.0789816 + 0.996876i \(0.525167\pi\)
\(858\) −1.43648e13 −0.904913
\(859\) −2.48059e13 −1.55448 −0.777242 0.629202i \(-0.783382\pi\)
−0.777242 + 0.629202i \(0.783382\pi\)
\(860\) −4.86214e12 −0.303099
\(861\) 2.61472e13 1.62148
\(862\) 4.86341e12 0.300026
\(863\) −1.49580e13 −0.917962 −0.458981 0.888446i \(-0.651785\pi\)
−0.458981 + 0.888446i \(0.651785\pi\)
\(864\) −1.94872e12 −0.118970
\(865\) −1.78345e13 −1.08315
\(866\) −2.08779e13 −1.26141
\(867\) 8.64771e12 0.519775
\(868\) −1.10995e13 −0.663690
\(869\) −1.19457e13 −0.710596
\(870\) 7.09070e12 0.419616
\(871\) −3.22790e12 −0.190037
\(872\) −2.85069e13 −1.66965
\(873\) −1.03652e13 −0.603969
\(874\) −2.08009e12 −0.120582
\(875\) −2.95608e13 −1.70482
\(876\) 1.63413e11 0.00937599
\(877\) −6.24566e12 −0.356517 −0.178258 0.983984i \(-0.557046\pi\)
−0.178258 + 0.983984i \(0.557046\pi\)
\(878\) 4.30854e12 0.244684
\(879\) 2.39300e12 0.135205
\(880\) −1.21105e13 −0.680756
\(881\) 2.33730e13 1.30714 0.653572 0.756864i \(-0.273270\pi\)
0.653572 + 0.756864i \(0.273270\pi\)
\(882\) 1.20803e13 0.672156
\(883\) 5.34930e12 0.296124 0.148062 0.988978i \(-0.452696\pi\)
0.148062 + 0.988978i \(0.452696\pi\)
\(884\) −3.17135e12 −0.174666
\(885\) −1.46515e12 −0.0802853
\(886\) 3.73661e12 0.203716
\(887\) −2.76665e12 −0.150071 −0.0750357 0.997181i \(-0.523907\pi\)
−0.0750357 + 0.997181i \(0.523907\pi\)
\(888\) 1.51423e13 0.817209
\(889\) −1.71414e13 −0.920424
\(890\) 1.99713e13 1.06697
\(891\) 2.32951e12 0.123827
\(892\) 4.41899e12 0.233712
\(893\) −8.75306e11 −0.0460605
\(894\) 1.76736e12 0.0925349
\(895\) 1.75966e13 0.916698
\(896\) 9.69240e12 0.502395
\(897\) 2.92676e13 1.50946
\(898\) −6.86636e12 −0.352357
\(899\) −1.78929e13 −0.913615
\(900\) −2.99021e11 −0.0151919
\(901\) −5.34042e12 −0.269969
\(902\) −2.75480e13 −1.38567
\(903\) −1.88004e13 −0.940960
\(904\) −7.54032e12 −0.375519
\(905\) 2.38078e13 1.17978
\(906\) −1.88753e12 −0.0930716
\(907\) −5.41718e12 −0.265791 −0.132896 0.991130i \(-0.542428\pi\)
−0.132896 + 0.991130i \(0.542428\pi\)
\(908\) 3.50704e12 0.171220
\(909\) 3.13908e12 0.152498
\(910\) 5.77337e13 2.79089
\(911\) −3.39211e13 −1.63169 −0.815844 0.578272i \(-0.803727\pi\)
−0.815844 + 0.578272i \(0.803727\pi\)
\(912\) 6.61384e11 0.0316575
\(913\) 2.42970e13 1.15727
\(914\) −2.55474e13 −1.21085
\(915\) −1.09991e13 −0.518754
\(916\) 6.40740e12 0.300713
\(917\) 7.76739e12 0.362755
\(918\) −1.07560e12 −0.0499874
\(919\) 3.66041e13 1.69282 0.846409 0.532534i \(-0.178760\pi\)
0.846409 + 0.532534i \(0.178760\pi\)
\(920\) 3.86313e13 1.77785
\(921\) 4.83347e11 0.0221356
\(922\) 8.94561e12 0.407682
\(923\) −5.44055e13 −2.46737
\(924\) 8.56782e12 0.386675
\(925\) 4.07911e12 0.183201
\(926\) 4.45695e12 0.199200
\(927\) 1.23449e13 0.549072
\(928\) −1.15541e13 −0.511409
\(929\) 1.54600e13 0.680989 0.340494 0.940247i \(-0.389406\pi\)
0.340494 + 0.940247i \(0.389406\pi\)
\(930\) −1.27789e13 −0.560168
\(931\) 5.38826e12 0.235058
\(932\) 4.54075e12 0.197131
\(933\) 2.18673e13 0.944775
\(934\) −2.27251e13 −0.977111
\(935\) 8.78480e12 0.375906
\(936\) −1.45695e13 −0.620447
\(937\) −3.37491e13 −1.43032 −0.715161 0.698960i \(-0.753646\pi\)
−0.715161 + 0.698960i \(0.753646\pi\)
\(938\) −4.02657e12 −0.169833
\(939\) 3.14772e12 0.132130
\(940\) 3.97321e12 0.165984
\(941\) −2.38639e12 −0.0992175 −0.0496087 0.998769i \(-0.515797\pi\)
−0.0496087 + 0.998769i \(0.515797\pi\)
\(942\) −9.25478e12 −0.382946
\(943\) 5.61277e13 2.31140
\(944\) −1.81660e12 −0.0744537
\(945\) −9.36255e12 −0.381901
\(946\) 1.98075e13 0.804120
\(947\) 1.40787e13 0.568837 0.284419 0.958700i \(-0.408199\pi\)
0.284419 + 0.958700i \(0.408199\pi\)
\(948\) −2.96131e12 −0.119082
\(949\) 2.14488e12 0.0858432
\(950\) 2.78943e11 0.0111112
\(951\) 1.81139e13 0.718124
\(952\) −1.61858e13 −0.638657
\(953\) 4.17831e12 0.164090 0.0820452 0.996629i \(-0.473855\pi\)
0.0820452 + 0.996629i \(0.473855\pi\)
\(954\) −5.99656e12 −0.234387
\(955\) −3.28541e13 −1.27813
\(956\) −1.19129e13 −0.461273
\(957\) 1.38117e13 0.532286
\(958\) 3.22945e13 1.23875
\(959\) −3.30626e13 −1.26227
\(960\) −1.75327e13 −0.666237
\(961\) 5.80701e12 0.219633
\(962\) 4.85774e13 1.82872
\(963\) 1.01847e13 0.381618
\(964\) −2.25753e12 −0.0841951
\(965\) −3.23161e13 −1.19963
\(966\) 3.65092e13 1.34898
\(967\) −2.15220e13 −0.791521 −0.395761 0.918354i \(-0.629519\pi\)
−0.395761 + 0.918354i \(0.629519\pi\)
\(968\) −7.19568e12 −0.263410
\(969\) −4.79758e11 −0.0174810
\(970\) −4.38908e13 −1.59184
\(971\) 3.09036e13 1.11564 0.557818 0.829963i \(-0.311639\pi\)
0.557818 + 0.829963i \(0.311639\pi\)
\(972\) 5.77478e11 0.0207509
\(973\) −9.50025e13 −3.39803
\(974\) −3.35823e13 −1.19563
\(975\) −3.92482e12 −0.139091
\(976\) −1.36375e13 −0.481073
\(977\) −4.38260e13 −1.53889 −0.769443 0.638715i \(-0.779466\pi\)
−0.769443 + 0.638715i \(0.779466\pi\)
\(978\) −1.57260e13 −0.549659
\(979\) 3.89014e13 1.35345
\(980\) −2.44585e13 −0.847056
\(981\) 1.48305e13 0.511265
\(982\) −3.38606e13 −1.16196
\(983\) 1.32092e13 0.451219 0.225609 0.974218i \(-0.427563\pi\)
0.225609 + 0.974218i \(0.427563\pi\)
\(984\) −2.79406e13 −0.950075
\(985\) 2.16996e13 0.734494
\(986\) −6.37730e12 −0.214877
\(987\) 1.53631e13 0.515291
\(988\) −1.58833e12 −0.0530316
\(989\) −4.03570e13 −1.34133
\(990\) 9.86413e12 0.326362
\(991\) −4.72326e13 −1.55565 −0.777823 0.628483i \(-0.783676\pi\)
−0.777823 + 0.628483i \(0.783676\pi\)
\(992\) 2.08227e13 0.682708
\(993\) −2.88847e13 −0.942748
\(994\) −6.78668e13 −2.20505
\(995\) −3.63894e13 −1.17699
\(996\) 6.02315e12 0.193935
\(997\) −1.97696e13 −0.633681 −0.316840 0.948479i \(-0.602622\pi\)
−0.316840 + 0.948479i \(0.602622\pi\)
\(998\) −5.95242e12 −0.189936
\(999\) −7.87769e12 −0.250239
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.10.a.c.1.15 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.c.1.15 22 1.1 even 1 trivial